OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..562
Eric Weisstein's World of Mathematics, Bell Polynomial.
FORMULA
a(n) = Sum_{k=0..floor(n/2)} 3^k * Stirling2(n,2*k).
a(n) = ( Bell_n(sqrt(3)) + Bell_n(-sqrt(3)) )/2, where Bell_n(x) is n-th Bell polynomial.
a(n) = 1; a(n) = 3 * Sum_{k=0..n-1} binomial(n-1, k) * A357572(k).
EXAMPLE
G.f. = 1 + 3*x^2 + 9*x^3 + 30*x^4 + 135*x^5 + 705*x^6 + ... - Michael Somos, Oct 06 2022
MATHEMATICA
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Cosh[Sqrt[3] * (Exp@x - 1)], {x, 0, n}]]; (* Michael Somos, Oct 06 2022 *)
PROG
(PARI) a(n) = sum(k=0, n\2, 3^k*stirling(n, 2*k, 2));
(PARI) my(x='x+O('x^30)); apply(round, Vec(serlaplace(cosh(sqrt(3) * (exp(x) - 1))))) \\ Michel Marcus, Oct 06 2022
(PARI) {a(n) = if(n<0, 0, n!*simplify(polcoeff( cosh(quadgen(12) * (exp(x + x*O(x^n)) - 1)), n)))}; /* Michael Somos, Oct 06 2022 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 06 2022
STATUS
approved